I fixed the problem myself. Unfortunately, it was even my own mistake, you couldn't have come up with the solution.
I don't access the Joomla site directly but there is an HTTP gateway before it. The problem was that Joomla sends two "Set-Cookie" headers in the Login-Response, but my gateway merged them and only sent the second one back to the browser. After ...
Under the Global Configuration settings menu item and the Site tab is there a value in the 'Cookie Settings' or 'Cookie Path' and if so do they reflect the domain of where you have copied the Joomla site.
Both fields can be blank and I think this is the recommendation these days.
1 - Move the site to the other host using Akeeba Backup, which is easier and you are less likely to make errors during the process.
2- Use PHP 7.2.x. Even that Joomla mention PHP 5 as minimum, PHP 7 is recommended and yes you need to update your site to the latest Joomla 3 version. Normally the customisations ...